Speaking of Set Theory

So, while I implied I would talk a little about Categories, since we just had our Set Theory final Exam (Saij and I), I thought perhaps a little discussion of this would be in order… Forgive the pun.

Indeed, while the basics of set theory were covered as always: looking at sets; their union and intersection; products of sets; and so forth, we also covered ordered sets. When order is dragged into the picture, we start to get a lot of interesting ideas that follow.

One very natural way of introducing order is to look at the ordering of sets by inclusion. Often, when one constructs… the natural numbers, say, one constructs them as sets of sets (of sets of sets, though usually not ad infinitum – in axiomatic set theory, the requirement that such chains be finite seems to come up a lot (though not always). So we can immediately look at an ordering of our construction by whether or not one thing is a subset of another.

The ordering by inclusion can also be used with sets that are otherwise ordered to show one thing or another… One example is to show that every partially ordered set which is not itself a chain contains (at least) two maximal chains. To show the existence of the first one, we look at the collection of all chains within the set and order it by inclusion. This gives us a lattice in which every chain… Here we need to be careful which chains we are talking about – not the chains from the original set, but chains in the ordered set whose elements are all chains in the first set (hopefully that is clear…). As I was saying, a lattice in which every chain has an upper bound – which by Zorn’s Lemma means the collection of chains (the lattice which we are talking about) has a maximal element!

I’ll leave with a teaser again- in my next post I’ll finish the proof and show from where the second maximal chain comes!

good night!

Felicis

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One Response

  1. Oooh, Let the Zornification process begin!

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